The value of
`"log"_(5) (1+ (1)/(5)) + "log"_(5) (1+(1)/(6)) + "log"_(5)(1+(1)/(7)) + …. + "log"_(5)(1+(1)/(624))` is

Options

A. 5
B. 4
C. 3
D. 2

Answer

To solve the expression

\[ \text{log}_{5} \left(1 + \frac{1}{5}\right) + \text{log}_{5} \left(1 + \frac{1}{6}\right) + \text{log}_{5} \left(1 + \frac{1}{7}\right) + \ldots + \text{log}_{5} \left(1 + \frac{1}{624}\right) \]

we will follow these steps:

### Step 1: Rewrite each logarithmic term
We start by rewriting each term in the series. The first term is:

\[ \text{log}_{5} \left(1 + \frac{1}{5}\right) = \text{log}_{5} \left(\frac{6}{5}\right) \]

The second term is:

\[ \text{log}_{5} \left(1 + \frac{1}{6}\right) = \text{log}_{5} \left(\frac{7}{6}\right) \]

The third term is:

\[ \text{log}_{5} \left(1 + \frac{1}{7}\right) = \text{log}_{5} \left(\frac{8}{7}\right) \]

Continuing this way, the last term is:

\[ \text{log}_{5} \left(1 + \frac{1}{624}\right) = \text{log}_{5} \left(\frac{625}{624}\right) \]

### Step 2: Combine the logarithmic terms
Using the property of logarithms that states \(\text{log}_{a}(x) + \text{log}_{a}(y) = \text{log}_{a}(xy)\), we can combine all the logarithmic terms:

\[ \text{log}_{5} \left(\frac{6}{5} \cdot \frac{7}{6} \cdot \frac{8}{7} \cdots \frac{625}{624}\right) \]

### Step 3: Simplify the product
Notice that in the product, all intermediate terms will cancel out:

\[ \frac{6}{5} \cdot \frac{7}{6} \cdot \frac{8}{7} \cdots \frac{625}{624} = \frac{625}{5} \]

### Step 4: Evaluate the logarithm
Now we can rewrite the expression as:

\[ \text{log}_{5} \left(\frac{625}{5}\right) = \text{log}_{5} (125) \]

Since \(125 = 5^3\), we have:

\[ \text{log}_{5} (125) = \text{log}_{5} (5^3) = 3 \]

### Step 5: Final result
Thus, the value of the original expression is:

\[ \boxed{3} \]